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Find the surface area of the figure. Round your answers to the nearest whole; if necessary: Leave your answers in terms of n for answers that contain T.20.8 f...

Question

Find the surface area of the figure. Round your answers to the nearest whole; if necessary: Leave your answers in terms of n for answers that contain T.20.8 f

Find the surface area of the figure. Round your answers to the nearest whole; if necessary: Leave your answers in terms of n for answers that contain T. 20.8 f



Answers

Find the surface area of the figure. Give both the exact value and an approximation to the nearest hundredth. (FIGURE CANNOT COPY)

So I am going to find the surface area of a cylinder and I have drawn my cylinder on. I'm feeling in a very colorful mood today. So that's why I have my glitter pen going on here. Um and so I do my cylinder. And then I also have the formula that I'm gonna use. So the surface area of a sliver is pi r squared plus two pi times, the radius times the hot. So now that I have my formula, I'm going to need Teoh put in o r identify the values that I'm gonna use in the formula for my variables. So my radius is actually given to me. Thankfully, it is six inches, and my height is also given to me. That's really bad as two inches. So now all I have to do is I just have to substitute these into my formula. So pi R Square. So that is a six and two pi R, which is a six times the height, which is a two. All right, so now all I'm gonna do is I'm gonna solve this. I will go ahead and I will solve it in a different color. That's a little less yellow. There's some bright stuff, but I know might be hard to read stuff. Um, there we go. It's still colorful, but a little easier. Right? So we'll have with the surface area of the cylinder is high are, uh, squared plus two. Hi. Are Oh, all right. So let's do order of operations here. Her speedy exponents. So we have high and Six Squared is 36 and Teoh And then we're gonna do multiplication. But I'm gonna leave the pie for right now. Um, so I'm gonna dio That's different. Do six times two is 12 and, uh, two times 12 is 24 So OK, so I have 36 pie plus 24 pi so I can combine my like terms and 36 close 24 is 60. Hi. So that's the exact answer. This is the surface area. Exactly. So in terms of pie square inches. But they also want me to round it to the hundreds, so I can say it is also approximately. I do 60 times pi, I get 100 it it and 50 hundreds inches square. So here are my values of the surface area

I am going, Teoh, find the surface area of this comb. And so to do that, I have drama cone. Angelos Wanna draw picture when you are dealing with geometry questions? And then I have written the formula for the surface area of a cone, which again you always want to write the formula of whatever you're going to be using to perform your operations. So all I have to do now is figure out what those values are that I'm gonna be using and then input them into my formula. So I need the radius because the surface area of a home is pi times the radius square I r squared plus high times, the radius times, the the slant height. And so the radius is not given to me. Sorry. Um, the slant height that that l is given to me, I have that That's nine feet, but I'm actually given the diameter all the way across the circle as three feet. But I know that the radius is half of the diameter, so I'm just going to take the diameter and divide it but you So that gives me one and 1/2 eight. And so now All I have to do is put that into my equation, that to substitute it. So everywhere I have a radius, I'm gonna put 1.5 and everywhere have a l for my slant height there and gonna have a line. And the last thing I'm going to do is solve So my surface area of my comb, the pie r squared plus high are so And then I'm just gonna do some order of operations here. So my first order of operation is gonna be to do my exponents of 1.5. So I'm gonna have high, and that's gonna be 2.25 plus hi. And I'm gonna multiply from left to right now. I'm gonna leave the pie in there because it does ask for an exact measurement and that's so pies and stay in there for for a minute. But so we have high. This isn't going to change this step, but 1.5 times nine is going to give me 13.5. And, um, I can combine my life terms. I have this orchid factor this out. I'm gonna factor my pie out so I'll get 2.25 plus 13.5, and I'll get 15.75 pi. So this is my exact answer. My exact answer for the surface area is 15 and 75 hundreds hi square feet, but they also want exact. So that means I'm going Teoh, multiply 15 and 75 hundreds by high and then round it to the hundreds. So that's going to give me 49 and 48 hundreds square feet, and that is there's a bunch of years after that. So that's a pretty easy round on that's the surface area for my come.

To find the surface area of this composite figure. Let's first start with a plan of how to go about making sure we hit every surface. We have a rectangular prism on the bottom, and on top of it sits a cylinder. So in order to include the entire surface, will need to find the surface area of the rectangular prism and the surface area of the cylinder. One thing to make a quick note off is when we find the area or the surface area of our cylinder. We're only going to use the area of one base because this bottom piece is not actually on the surface. The cylinder is sitting on top of the rectangular prism so that bottom base is no longer part of the total surface area. So we'll start by identifying what we're going to do. We're going to find the total surface area equal to the surface area of the rectangular prism, plus the surface area of the cylinder, but only one of its bases. So the surface area of the cylinder, minus one of its bases. When we read our formulas for this, we can account for this subtraction of the area of one of the bases of the cylinder within the variables that we're going to use. So let's start with a surface area of the rectangular prism that is going to be equal to. We're going to start with the prison here. Surface area is equal to the lateral area, plus the area of two of its bases. The lateral area well, that's equal to its perimeter times its height. Plus to be we know that the perimeter of the cylinder Sorry, The perimeter of the prism is going to be 14 plus eight plus 14 plus eight, and in this case had it all up. We get 44 and were given the height Have 12. Substitute those values in 44 times 12 plus two times. The area of the base area of the base is equal to the length times the width, which is 14 times a. We need two of those so into 14 times eight, which is 112 and then we'll simplify. So the surface area of the prism is equal to 528 plus 224 which is 752 square feet. They're gonna hold on to that over here for a sec. They were gonna move on to the surface area of the cylinder, were given a radius of four and a height of eight. Now, keep in mind we're going to subtract. One of the base is one of the areas of the base because it's not part of the surface area. Surface area of a cylinder is equal to two pi r age plus two pi r squared because typically it's twice the area of the base. Since we're only using one, we're gonna use pi r squared where given our is four h is eight. Substitute those values and then simplify. Two times four is eight times eight is 64 pi plus 16 pie and we have 80 pie b squared. Now we need the total surface areas. We're going to add these to individual figures together. Total surface area is equal to the surface area of the prism plus the surface area of the cylinder. And I forgot my pie. Let's put that in there before I forget simplifying this and we're going to round to the nearest 10th. So we have 752 plus 80 pie and that is approximately 1003.3. Don't forget your units. Square feet Total surface area 1003.3 square feet.

To find the surface area of this composite figure. Let's think about one exactly at his. We have a solid cylinder, and inside it is a block. We want to remove that block. So imagine you're looking down a tube. This will give you an idea of what the surface area is gonna look like. You're looking down the tube. It's a circle, which would be the top of the cylinder, and inside it we have a rectangular block. So the surface area we're looking at is going to be the entire surface outside, which would be your cylinder, plus the surface that remains on the inside upon removing that block. So we need to have this part of our rectangular prism accounted for, and that's going to be its lateral area. The lateral area of the bloc would be all of its lateral surfaces Here inside, we also need to account for when we remove that block, it's leaving a hole in the cylinder, so we need to subtract out the area of the bases from our block. So when we're writing our formulas, we're going to have the entire surface area of the cylinder. Add to it. The lateral area of our rectangular prism and then subtract from it the base areas of the rectangular prism. So total surface area is the surface area of the cylinder, added to it the lateral area of the prism and subtract the base area of the prism, which would be are to be the prison. So that's our plan. Let's start with our cylinder. Surface area of a cylinder is equal to two pi h r plus two pi r squared. Now remember, this is the entire surface area. We know that H is 14 that was given an R was also given at 14. Substituting these values we have. Our surface area is equal to two times pi times 14 times 14 plus two pi 14 squared. So this is going to simplify to 14 squared times two is 392 pi, and again we have 14 squared times to 392 pi Roman numbers a little backwards there. Total surface area of our cylinder is 784 pi R units or feed. So let's not forget those and then hold on to that value for a little bit. We're gonna move over to our prism. To find the lateral area of the prism, we need the perimeter of the base times its height. While the perimeter is twice the length plus twice the width, we know the length and the width those air given If we look at her diagram here, whips we have a length of looks like 14 and a width of six were also given the height. It's the exact same height is our cylinder. So age is equal to 14 substituting these values two times 14 plus two times six our perimeter times the height. So the perimeter is 28 12 which is 40 times 14. Is our height total surface or sorry. Total lateral area of our printable. Our prism is going to be 560 square feet and just one more thing to account for. And that is the subtraction of the base area of our prism. That inside part, that we're pulling out because now we've created a whole. So the base of the prism is going to be twice the length times the wits as before. Length is 14 with this six. So we're gonna have our base is equal to 14 lips two times 14 times. Six. When we simplify, this is 168 square feet, and then we need to put it all together. Total surface area is the surface area of our cylinder 784 pie, plus the lateral area of our rectangular prism, minus the base area of the rectangular prisms. Using our calculator 784 pi plus 560 minus 168 it's approximately 200 e. 2855 0.0 square feet.


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